Bricks over preprojective algebras and join-irreducible elements in Coxeter groups

Abstract

A (semi)brick over an algebra $A$ is a module $S$ such that the endomorphism ring ${\mathrm{End}}_A(S)$ is a (product of) division algebra. For each Dynkin diagram $\Delta$, there is a bijection from the Coxeter group $W$ of type $\Delta$ to the set of semibricks over the preprojective algebra $\Pi$ of type $\Delta$, which is restricted to a bijection from the set of join-irreducible elements of $W$ to the set of bricks over $\Pi$. This paper is devoted to giving an explicit description of these bijections in the case $\Delta ={\mathbb{A}}_n$ or ${\mathbb{D}}_n$. First, for each join-irreducible element $w\in W$, we describe the corresponding brick $S(w)$ in terms of "Young diagram-like" notation. Next, we determine the canonical join representation $w={\bigvee }_{i=1}^m{w}_i$ of an arbitrary element $w\in W$ based on Reading's work, and prove that ${⨁}_{i=1}^{n}S(w_i)$ is the semibrick corresponding to $w$.

37 pages

References in corpus (2)

• Minimal inclusions of torsion classes
• Derived Picard groups of preprojective algebras of Dynkin type

Cited by in corpus (3)

• Silting objects
• Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras
• Pairwise Compatibility for 2-Simple Minded Collections II: Preprojective Algebras and Semibrick Pairs of Full Rank